Three Applications of a Bonus Relation for Gravity Amplitudes
Marcus Spradlin, Anastasia Volovich, Congkao Wen
TL;DR
Arkani-Hamed et al. showed that all tree-level ${\cal N}=8$ SUGRA amplitudes fall off like $1/z^2$ under a special complex supermomentum shift, yielding bonus relations. This paper exploits those bonus relations for MHV amplitudes to convert $(n-2)!$-term, recursion-based expressions into $(n-3)!$-term BGK-type forms and demonstrates this with three concrete applications. It provides a direct proof of Elvang–Freedman’s EF2 formula, derives a BBST-based simplified expression, and offers an alternate Mason–Skinner proof, thereby also showing that the BGK formula satisfies the on-shell recursion. Overall, the work clarifies the extra structure in tree-level graviton amplitudes and highlights the power of bonus relations to simplify and relate different MHV representations.
Abstract
Arkani-Hamed et. al. have recently shown that all tree-level scattering amplitudes in maximal supergravity exhibit exceptionally soft behavior when two supermomenta are taken to infinity in a particular complex direction, and that this behavior implies new non-trivial relations amongst amplitudes in addition to the well-known on-shell recursion relations. We consider the application of these new bonus relations to MHV amplitudes, showing that they can be used quite generally to relate (n-2)!-term formulas typically obtained from recursion relations to (n-3)!-term formulas related to the original BGK conjecture. Specifically we provide (1) a direct proof of a formula presented by Elvang and Freedman, (2) a new formula based on one due to Bedford et. al., and (3) an alternate proof of a formula recently obtained by Mason and Skinner. Our results also provide the first direct proof that the conjectured BGK formula, only very recently proven via completely different methods, satisfies the on-shell recursion.